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Minkowski space

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Minkowski_space

In physics, Minkowski space (or Minkowski spacetime) (/mɪŋˈkɔːfski, -ˈkɒf-/) is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.

The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others said it "was grown on experimental physical grounds".

Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. Minkowski space differs from four-dimensional Euclidean space insofar as it treats time differently than the three spatial dimensions.

In 3-dimensional Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by rotations, reflections and translations. When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations is called the Poincaré group. Minkowski's model follows special relativity, where motion causes time dilation changing the scale applied to the frame in motion and shifts the phase of light.

Spacetime is equipped with an indefinite non-degenerate bilinear form, called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as an argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The group of transformations for Minkowski space that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is the Poincaré group (as opposed to the Galilean group).

History

Complex Minkowski spacetime

In his second relativity paper in 1905, Henri Poincaré showed how, by taking time to be an imaginary fourth spacetime coordinate $ict$ , where $c$ is the speed of light and $i$ is the imaginary unit, Lorentz transformations can be visualized as ordinary rotations of the four-dimensional Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each point representing an event in spacetime. The Lorentz transformations can then be thought of as rotations in this four-dimensional space, where the rotation axis corresponds to the direction of relative motion between the two observers and the rotation angle is related to their relative velocity.

To understand this concept, one should consider the coordinates of an event in spacetime represented as a four-vector $(t, x, y, z)$ . A Lorentz transformation is represented by a matrix that acts on the four-vector, changing its components. This matrix can be thought of as a rotation matrix in four-dimensional space, which rotates the four-vector around a particular axis. $x^{2} + y^{2} + z^{2} + (i c t)^{2} = constant .$

Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as Euclidean rotations and are interpreted in the ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with real inertial coordinates. The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary, which turns rotations into rotations in hyperbolic space (see hyperbolic rotation).

This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in German published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". He reformulated Maxwell equations as a symmetrical set of equations in the four variables $(x, y, z, ict)$ combined with redefined vector variables for electromagnetic quantities, and he was able to show directly and very simply their invariance under Lorentz transformation. He also made other important contributions and used matrix notation for the first time in this context. From his reformulation, he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum.

Real Minkowski spacetime

In a further development in his 1908 "Space and Time" lecture, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables $(x, y, z, t)$ of space and time in the coordinate form in a four-dimensional real vector space. Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, and events not on the light cone are classified by their relation to the apex as spacelike or timelike. It is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity.

In the English translation of Minkowski's paper, the Minkowski metric, as defined below, is referred to as the line element. The Minkowski inner product below appears unnamed when referring to orthogonality (which he calls normality) of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is a translation dependent) as "sum".

Minkowski's principal tool is the Minkowski diagram, and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g., proper time and length contraction) and to provide geometrical interpretation to the generalization of Newtonian mechanics to relativistic mechanics. For these special topics, see the referenced articles, as the presentation below will be principally confined to the mathematical structure (Minkowski metric and from it derived quantities and the Poincaré group as symmetry group of spacetime) following from the invariance of the spacetime interval on the spacetime manifold as consequences of the postulates of special relativity, not to specific application or derivation of the invariance of the spacetime interval. This structure provides the background setting of all present relativistic theories, barring general relativity for which flat Minkowski spacetime still provides a springboard as curved spacetime is locally Lorentzian.

Minkowski, aware of the fundamental restatement of the theory which he had made, said

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
— Hermann Minkowski, 1908, 1909

Though Minkowski took an important step for physics, Albert Einstein saw its limitation:

At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a quasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of gravitation. He was still far from the study of curvilinear coordinates and Riemannian geometry, and the heavy mathematical apparatus entailed.

For further historical information see references Galison (1979), Corry (1997) and Walter (1999).

Causal structure

Where $v$ is velocity, $x$ , $y$ , and $z$ are Cartesian coordinates in 3-dimensional space, $c$ is the constant representing the universal speed limit, and $t$ is time, the four-dimensional vector $v = (ct, x, y, z) = (ct, r)$ is classified according to the sign of $c 2 t 2 - r 2$ . A vector is timelike if $c 2 t 2 > r 2$ , spacelike if $c 2 t 2 < r 2$ , and null or lightlike if $c 2 t 2 = r 2$ . This can be expressed in terms of the sign of $η (v, v)$ , also called scalar product, as well, which depends on the signature. The classification of any vector will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the spacetime interval under Lorentz transformation.

The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. Given a timelike vector $v$ , there is a worldline of constant velocity associated with it, represented by a straight line in a Minkowski diagram.

Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has

future-directed timelike vectors whose first component is positive (tip of vector located in causal future (also called the absolute future) in the figure) and
past-directed timelike vectors whose first component is negative (causal past (also called the absolute past)).

Null vectors fall into three classes:

the zero vector, whose components in any basis are $(0, 0, 0, 0)$ (origin),
future-directed null vectors whose first component is positive (upper light cone), and
past-directed null vectors whose first component is negative (lower light cone).

Together with spacelike vectors, there are 6 classes in all.

An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases, it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis.

Vector fields are called timelike, spacelike, or null if the associated vectors are timelike, spacelike, or null at each point where the field is defined.

Properties of time-like vectors

Time-like vectors have special importance in the theory of relativity as they correspond to events that are accessible to the observer at (0, 0, 0, 0) with a speed less than that of light. Of most interest are time-like vectors that are similarly directed, i.e. all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise because both forward and backward cones are convex, whereas the space-like region is not convex.

Scalar product

The scalar product of two time-like vectors $u 1 = (t 1, x 1, y 1, z 1)$ and $u 2 = (t 2, x 2, y 2, z 2)$ is $η (u_{1}, u_{2}) = u_{1} \cdot u_{2} = c^{2} t_{1} t_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2} .$

Positivity of scalar product: An important property is that the scalar product of two similarly directed time-like vectors is always positive. This can be seen from the reversed Cauchy–Schwarz inequality below. It follows that if the scalar product of two vectors is zero, then one of these, at least, must be space-like. The scalar product of two space-like vectors can be positive or negative as can be seen by considering the product of two space-like vectors having orthogonal spatial components and times either of different or the same signs.

Using the positivity property of time-like vectors, it is easy to verify that a linear sum with positive coefficients of similarly directed time-like vectors is also similarly directed time-like (the sum remains within the light cone because of convexity).

Norm and reversed Cauchy inequality

The norm of a time-like vector $u = (ct, x, y, z)$ is defined as $‖ u ‖ = \sqrt{η (u, u)} = \sqrt{c^{2} t^{2} - x^{2} - y^{2} - z^{2}}$

The reversed Cauchy inequality is another consequence of the convexity of either light cone. For two distinct similarly directed time-like vectors $u 1$ and $u 2$ this inequality is $η (u_{1}, u_{2}) > ‖ u_{1} ‖ ‖ u_{2} ‖$ or algebraically, $c^{2} t_{1} t_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2} > \sqrt{(c^{2} t_{1}^{2} - x_{1}^{2} - y_{1}^{2} - z_{1}^{2}) (c^{2} t_{2}^{2} - x_{2}^{2} - y_{2}^{2} - z_{2}^{2})}$

From this, the positive property of the scalar product can be seen.

Reversed triangle inequality

For two similarly directed time-like vectors $u$ and $w$ , the inequality is $‖ u + w ‖ \geq ‖ u ‖ + ‖ w ‖,$ where the equality holds when the vectors are linearly dependent.

The proof uses the algebraic definition with the reversed Cauchy inequality: $\begin{aligned} {‖ u + w ‖}^{2} & = {‖ u ‖}^{2} + 2 (u, w) + {‖ w ‖}^{2} \\ \geq {‖ u ‖}^{2} + 2 ‖ u ‖ ‖ w ‖ + {‖ w ‖}^{2} = {(‖ u ‖ + ‖ w ‖)}^{2} . \end{aligned}$

The result now follows by taking the square root on both sides.

Mathematical structure

It is assumed below that spacetime is endowed with a coordinate system corresponding to an inertial frame. This provides an origin, which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to an affine space can remove the extra structure. However, this is not the introductory convention and is not covered here.

For an overview, Minkowski space is a $4$ -dimensional real vector space equipped with a non-degenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the Minkowski inner product, with metric signature either $(+ - - -)$ or $(- + + +)$ . The tangent space at each event is a vector space of the same dimension as spacetime, $4$ .

Tangent vectors

In practice, one need not be concerned with the tangent spaces. The vector space structure of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. Lee (2003, Proposition 3.8.) or Lee (2012, Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as $\begin{aligned} (x^{0}, x^{1}, x^{2}, x^{3}) & \leftrightarrow {x^{0} e_{0} |}_{p} + {x^{1} e_{1} |}_{p} + {x^{2} e_{2} |}_{p} + {x^{3} e_{3} |}_{p} \\ \leftrightarrow {x^{0} e_{0} |}_{q} + {x^{1} e_{1} |}_{q} + {x^{2} e_{2} |}_{q} + {x^{3} e_{3} |}_{q} \end{aligned}$ with basis vectors in the tangent spaces defined by ${e_{μ} |}_{p} = {\frac{\partial}{\partial x^{μ}} |}_{p} or e_{0} |_{p} = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}), etc .$

Here, $p$ and $q$ are any two events, and the second basis vector identification is referred to as parallel transport. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first-order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a directional derivative operator on the set of smooth functions. This is promoted to a definition of tangent vectors in manifolds not necessarily being embedded in $R n$ . This definition of tangent vectors is not the only possible one, as ordinary n-tuples can be used as well.

For some purposes, it is desirable to identify tangent vectors at a point $p$ with displacement vectors at $p$ , which is, of course, admissible by essentially the same canonical identification. The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in Misner, Thorne & Wheeler (1973). They offer various degrees of sophistication (and rigor) depending on which part of the material one chooses to read.

Metric signature

The metric signature refers to which sign the Minkowski inner product yields when given space (spacelike to be specific, defined further down) and time basis vectors (timelike) as arguments. Further discussion about this theoretically inconsequential but practically necessary choice for purposes of internal consistency and convenience is deferred to the hide box below. See also the page treating sign convention in Relativity.

Terminology

Mathematically associated with the bilinear form is a tensor of type $(0,2)$ at each point in spacetime, called the Minkowski metric. The Minkowski metric, the bilinear form, and the Minkowski inner product are all the same object; it is a bilinear function that accepts two (contravariant) vectors and returns a real number. In coordinates, this is the $4\times4$ matrix representing the bilinear form.

For comparison, in general relativity, a Lorentzian manifold $L$ is likewise equipped with a metric tensor $g$ , which is a nondegenerate symmetric bilinear form on the tangent space $T p L$ at each point $p$ of $L$ . In coordinates, it may be represented by a $4\times4$ matrix depending on spacetime position. Minkowski space is thus a comparatively simple special case of a Lorentzian manifold. Its metric tensor is in coordinates with the same symmetric matrix at every point of $M$ , and its arguments can, per above, be taken as vectors in spacetime itself.

Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension $n = 4$ and signature $(1, 3)$ or $(3, 1)$ . Elements of Minkowski space are called events. Minkowski space is often denoted $R 1,3$ or $R 3,1$ to emphasize the chosen signature, or just $M$ . It is an example of a pseudo-Riemannian manifold.

Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space $V$ , that is, $η : V \times V \to R$ where $η$ has signature $(+, -, -, -)$ , and signature is a coordinate-invariant property of $η$ . The space of bilinear maps forms a vector space which can be identified with $M^{*} \otimes M^{*}$ , and $η$ may be equivalently viewed as an element of this space. By making a choice of orthonormal basis ${e_{μ}}$ , $M := (V, η)$ can be identified with the space $R^{1, 3} := (R^{4}, η_{μ ν})$ . The notation is meant to emphasize the fact that $M$ and $R^{1, 3}$ are not just vector spaces but have added structure. $η_{μ ν} = diag (+ 1, - 1, - 1, - 1)$ .

An interesting example of non-inertial coordinates for (part of) Minkowski spacetime is the Born coordinates. Another useful set of coordinates is the light-cone coordinates.

Pseudo-Euclidean metrics

The Minkowski inner product is not an inner product, since it is not positive-definite, i.e. the quadratic form $η (v, v)$ need not be positive for nonzero $v$ . The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be indefinite. The Minkowski metric $η$ is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a constant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type $(0, 2)$ tensor. It accepts two arguments $u p, v p$ , vectors in $T p M, p \in M$ , the tangent space at $p$ in $M$ . Due to the above-mentioned canonical identification of $T p M$ with $M$ itself, it accepts arguments $u, v$ with both $u$ and $v$ in $M$ .

As a notational convention, vectors $v$ in $M$ , called 4-vectors, are denoted in italics, and not, as is common in the Euclidean setting, with boldface $v$ . The latter is generally reserved for the $3$ -vector part (to be introduced below) of a $4$ -vector.

The definition $u \cdot v = η (u, v)$ yields an inner product-like structure on $M$ , previously and also henceforth, called the Minkowski inner product, similar to the Euclidean inner product, but it describes a different geometry. It is also called the relativistic dot product. If the two arguments are the same, $u \cdot u = η (u, u) \equiv ‖ u ‖^{2} \equiv u^{2},$ the resulting quantity will be called the Minkowski norm squared. The Minkowski inner product satisfies the following properties.

Linearity in the first argument: $η (a u + v, w) = a η (u, w) + η (v, w), \forall u, v \in M, \forall a \in R$
Symmetry: $η (u, v) = η (v, u)$
Non-degeneracy: $η (u, v) = 0, \forall v \in M \Rightarrow u = 0$

The first two conditions imply bilinearity. The defining difference between a pseudo-inner product and an inner product proper is that the former is not required to be positive definite, that is, $η (u, u) < 0$ is allowed.

The most important feature of the inner product and norm squared is that these are quantities unaffected by Lorentz transformations. In fact, it can be taken as the defining property of a Lorentz transformation in that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for all classical groups definable this way in classical group. There, the matrix $Φ$ is identical in the case $O(3, 1)$ (the Lorentz group) to the matrix $η$ to be displayed below.

Two vectors $v$ and $w$ are said to be orthogonal if $η (v, w) = 0$ . For a geometric interpretation of orthogonality in the special case, when $η (v, v) \leq 0$ and $η (w, w) \geq 0$ (or vice versa), see hyperbolic orthogonality.

A vector $e$ is called a unit vector if $η (e, e) = \pm1$ . A basis for $M$ consisting of mutually orthogonal unit vectors is called an orthonormal basis.

For a given inertial frame, an orthonormal basis in space, combined with the unit time vector, forms an orthonormal basis in Minkowski space. The number of positive and negative unit vectors in any such basis is a fixed pair of numbers equal to the signature of the bilinear form associated with the inner product. This is Sylvester's law of inertia.

More terminology (but not more structure): The Minkowski metric is a pseudo-Riemannian metric, more specifically, a Lorentzian metric, even more specifically, the Lorentz metric, reserved for $4$ -dimensional flat spacetime with the remaining ambiguity only being the signature convention.

Minkowski metric

From the second postulate of special relativity, together with homogeneity of spacetime and isotropy of space, it follows that the spacetime interval between two arbitrary events called $1$ and $2$ is: $c^{2} {(t_{1} - t_{2})}^{2} - {(x_{1} - x_{2})}^{2} - {(y_{1} - y_{2})}^{2} - {(z_{1} - z_{2})}^{2} .$ This quantity is not consistently named in the literature. The interval is sometimes referred to as the square root of the interval as defined here.

The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of $c^{2} t^{2} - x^{2} - y^{2} - z^{2}$ provided the transformations are linear. This quadratic form can be used to define a bilinear form $u \cdot v = c^{2} t_{1} t_{2} - x_{1} x_{2} - y_{1} y_{2} - z_{1} z_{2}$ via the polarization identity. This bilinear form can in turn be written as $u \cdot v = u^{T} [η] v,$ where $[η]$ is a $4 \times 4$ matrix associated with $η$ . While possibly confusing, it is common practice to denote $[η]$ with just $η$ . The matrix is read off from the explicit bilinear form as $η = (\begin{array}{r} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}),$ and the bilinear form $u \cdot v = η (u, v),$ with which this section started by assuming its existence, is now identified.

For definiteness and shorter presentation, the signature $(- + + +)$ is adopted below. This choice (or the other possible choice) has no (known) physical implications. The symmetry group preserving the bilinear form with one choice of signature is isomorphic (under the map given here) with the symmetry group preserving the other choice of signature. This means that both choices are in accord with the two postulates of relativity. Switching between the two conventions is straightforward. If the metric tensor $η$ has been used in a derivation, go back to the earliest point where it was used, substitute $η$ for $- η$ , and retrace forward to the desired formula with the desired metric signature.

Standard basis

A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors ${e 0, e 1, e 2, e 3}$ such that $η (e_{0}, e_{0}) = - η (e_{1}, e_{1}) = - η (e_{2}, e_{2}) = - η (e_{3}, e_{3}) = 1$ and for which $η (e_{μ}, e_{ν}) = 0$ when $μ \neq ν .$

These conditions can be written compactly in the form $η (e_{μ}, e_{ν}) = η_{μ ν} .$

Relative to a standard basis, the components of a vector $v$ are written $(v 0, v 1, v 2, v 3)$ where the Einstein notation is used to write $v = v μ e μ$ . The component $v 0$ is called the timelike component of $v$ while the other three components are called the spatial components. The spatial components of a $4$ -vector $v$ may be identified with a $3$ -vector $v = (v 1, v 2, v 3)$ .

In terms of components, the Minkowski inner product between two vectors $v$ and $w$ is given by

$η (v, w) = η_{μ ν} v^{μ} w^{ν} = v^{0} w_{0} + v^{1} w_{1} + v^{2} w_{2} + v^{3} w_{3} = v^{μ} w_{μ} = v_{μ} w^{μ},$ and $η (v, v) = η_{μ ν} v^{μ} v^{ν} = v^{0} v_{0} + v^{1} v_{1} + v^{2} v_{2} + v^{3} v_{3} = v^{μ} v_{μ} .$

Here lowering of an index with the metric was used.

There are many possible choices of standard basis obeying the condition $η (e_{μ}, e_{ν}) = η_{μ ν} .$ Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix $Λ_{ν}^{μ}$ , a real $4 \times 4$ matrix satisfying $Λ_{ρ}^{μ} η_{μ ν} Λ_{σ}^{ν} = η_{ρ σ} .$ or $Λ$ , a linear map on the abstract vector space satisfying, for any pair of vectors $u$ , $v$ , $η (Λ u, Λ v) = η (u, v) .$

Then if two different bases exist, ${e 0, e 1, e 2, e 3}$ and ${e' 0, e' 1, e' 2, e' 3}$ , $e_{μ}^{'} = e_{ν} Λ_{μ}^{ν}$ can be represented as $e_{μ}^{'} = e_{ν} Λ_{μ}^{ν}$ or $e_{μ}^{'} = Λ e_{μ}$ . While it might be tempting to think of $Λ_{ν}^{μ}$ and $Λ$ as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.

Raising and lowering of indices

Technically, a non-degenerate bilinear form provides a map between a vector space and its dual; in this context, the map is between the tangent spaces of $M$ and the cotangent spaces of $M$ . At a point in $M$ , the tangent and cotangent spaces are dual vector spaces (so the dimension of the cotangent space at an event is also $4$ ). Just as an authentic inner product on a vector space with one argument fixed, by Riesz representation theorem, may be expressed as the action of a linear functional on the vector space, the same holds for the Minkowski inner product of Minkowski space.

Thus if $v μ$ are the components of a vector in tangent space, then $η μν v μ = v ν$ are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors in $M$ itself, this is mostly ignored, and vectors with lower indices are referred to as covariant vectors. In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices are contravariant vectors. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of $η$ in matrix representation, can be used to define raising of an index. The components of this inverse are denoted $η μν$ . It happens that $η μν = η μν$ . These maps between a vector space and its dual can be denoted $η ♭$ (eta-flat) and $η ♯$ (eta-sharp) by the musical analogy.

Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear function can be characterized by two objects: its kernel, which is a hyperplane passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing depending on the norm (bigger = smaller spacing), with one of them (the kernel) passing through the origin. The mathematical term for a covariant vector is 1-covector or 1-form (though the latter is usually reserved for covector fields).

One quantum mechanical analogy explored in the literature is that of a de Broglie wave (scaled by a factor of Planck's reduced constant) associated with a momentum four-vector to illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the action of the covariant version of one of them on the contravariant version of the other. The inner product is then how many times the arrow pierces the planes. The mathematical reference, Lee (2003), offers the same geometrical view of these objects (but mentions no piercing).

The electromagnetic field tensor is a differential 2-form, which geometrical description can as well be found in MTW.

One may, of course, ignore geometrical views altogether (as is the style in e.g. Weinberg (2002) and Landau & Lifshitz 2002) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to as index gymnastics, ensures that moving vectors around and changing from contravariant to covariant vectors and vice versa (as well as higher order tensors) is mathematically sound. Incorrect expressions tend to reveal themselves quickly.

Coordinate free raising and lowering

Given a bilinear form $η : M \times M \to R$ , the lowered version of a vector can be thought of as the partial evaluation of $η$ , that is, there is an associated partial evaluation map $η (\cdot, -) : M \to M^{*}; v \mapsto η (v, \cdot) .$

The lowered vector $η (v, \cdot) \in M^{*}$ is then the dual map $u \mapsto η (v, u)$ . Note it does not matter which argument is partially evaluated due to the symmetry of $η$ .

Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy indicates that the kernel of the map is trivial. In finite dimension, as is the case here, and noting that the dimension of a finite-dimensional space is equal to the dimension of the dual, this is enough to conclude the partial evaluation map is a linear isomorphism from $M$ to $M^{*}$ . This then allows the definition of the inverse partial evaluation map, $η^{- 1} : M^{*} \to M,$ which allows the inverse metric to be defined as $η^{- 1} : M^{*} \times M^{*} \to R, η^{- 1} (α, β) = η (η^{- 1} (α), η^{- 1} (β))$ where the two different usages of $η^{- 1}$ can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metric $η -1$ is indeed the matrix inverse to $η$ .

Formalism of the Minkowski metric

The present purpose is to show semi-rigorously how formally one may apply the Minkowski metric to two vectors and obtain a real number, i.e. to display the role of the differentials and how they disappear in a calculation. The setting is that of smooth manifold theory, and concepts such as convector fields and exterior derivatives are introduced.

Chronological and causality relations

Let $x, y \in M$ . Here,

$x$ chronologically precedes $y$ if $y - x$ is future-directed timelike. This relation has the transitive property and so can be written $x < y$ .
$x$ causally precedes $y$ if $y - x$ is future-directed null or future-directed timelike. It gives a partial ordering of spacetime and so can be written $x \leq y$ .

Suppose $x \in M$ is timelike. Then the simultaneous hyperplane for $x$ is ${y : η (x, y) = 0}$ . Since this hyperplane varies as $x$ varies, there is a relativity of simultaneity in Minkowski space.

Generalizations

A Lorentzian manifold is a generalization of Minkowski space in two ways. The total number of spacetime dimensions is not restricted to be $4$ ( $2$ or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.

Complexified Minkowski space

Complexified Minkowski space is defined as $M c = M \oplus iM$ . Its real part is the Minkowski space of four-vectors, such as the four-velocity and the four-momentum, which are independent of the choice of orientation of the space. The imaginary part, on the other hand, may consist of four pseudovectors, such as angular velocity and magnetic moment, which change their direction with a change of orientation. A pseudoscalar $i$ is introduced, which also changes sign with a change of orientation. Thus, elements of $M c$ are independent of the choice of the orientation.

The inner product-like structure on $M c$ is defined as $u \cdot v = η (u, v)$ for any $u, v \in M c$ . A relativistic pure spin of an electron or any half spin particle is described by $ρ \in M c$ as $ρ = u + is$ , where $u$ is the four-velocity of the particle, satisfying $u 2 = 1$ and $s$ is the 4D spin vector, which is also the Pauli–Lubanski pseudovector satisfying $s 2 = -1$ and $u \cdot s = 0$ .

Generalized Minkowski space

Minkowski space refers to a mathematical formulation in four dimensions. However, the mathematics can easily be extended or simplified to create an analogous generalized Minkowski space in any number of dimensions. If $n \geq 2$ , $n$ -dimensional Minkowski space is a vector space of real dimension $n$ on which there is a constant Minkowski metric of signature $(n - 1, 1)$ or $(1, n - 1)$ . These generalizations are used in theories where spacetime is assumed to have more or less than $4$ dimensions. String theory and M-theory are two examples where $n > 4$ . In string theory, there appears conformal field theories with $1 + 1$ spacetime dimensions.

de Sitter space can be formulated as a submanifold of generalized Minkowski space as can the model spaces of hyperbolic geometry (see below).

Curvature

As a flat spacetime, the three spatial components of Minkowski spacetime always obey the Pythagorean Theorem. Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significant gravitation. However, in order to take gravity into account, physicists use the theory of general relativity, which is formulated in the mathematics of a non-Euclidean geometry. When this geometry is used as a model of physical space, it is known as curved space.

Even in curved space, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities). More abstractly, it can be said that in the presence of gravity spacetime is described by a curved 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.

Geometry

The meaning of the term geometry for the Minkowski space depends heavily on the context. Minkowski space is not endowed with Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the model spaces in hyperbolic geometry (negative curvature) and the geometry modeled by the sphere (positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not a metric space and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains submanifolds endowed with a Riemannian metric yielding hyperbolic geometry.

Model spaces of hyperbolic geometry of low dimension, say 2 or 3, cannot be isometrically embedded in Euclidean space with one more dimension, i.e. $R^{3}$ or $R^{4}$ respectively, with the Euclidean metric $\bar{g}$ , preventing easy visualization. By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension. Hyperbolic spaces can be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric $η$ .

Define $H_{R}^{1 (n)} \subset M^{n + 1}$ to be the upper sheet ( $c t > 0$ ) of the hyperboloid $H_{R}^{1 (n)} = {(c t, x^{1}, \dots, x^{n}) \in M^{n} : c^{2} t^{2} - {(x^{1})}^{2} - \dots - {(x^{n})}^{2} = R^{2}, c t > 0}$ in generalized Minkowski space $M^{n + 1}$ of spacetime dimension $n + 1.$ This is one of the surfaces of transitivity of the generalized Lorentz group. The induced metric on this submanifold, $h_{R}^{1 (n)} = ι^{*} η,$ the pullback of the Minkowski metric $η$ under inclusion, is a Riemannian metric. With this metric $H_{R}^{1 (n)}$ is a Riemannian manifold. It is one of the model spaces of Riemannian geometry, the hyperboloid model of hyperbolic space. It is a space of constant negative curvature $- 1 / R^{2}$ .^[26] The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the $n$ for its dimension. A $2 (2)$ corresponds to the Poincaré disk model, while $3 (n)$ corresponds to the Poincaré half-space model of dimension $n .$

Preliminaries

In the definition above $ι : H_{R}^{1 (n)} \to M^{n + 1}$ is the inclusion map and the superscript star denotes the pullback. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that $H_{R}^{1 (n)}$ actually is a hyperbolic space.

Behavior of tensors under inclusion, pullback of covariant tensors under general maps and pushforward of vectors under general maps

Hyperbolic stereographic projection

Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.

In order to exhibit the metric, it is necessary to pull it back via a suitable parametrization. A parametrization of a submanifold $S$ of a manifold $M$ is a map $U \subset R m \to M$ whose range is an open subset of $S$ . If $S$ has the same dimension as $M$ , a parametrization is just the inverse of a coordinate map $φ : M \to U \subset R m$ . The parametrization to be used is the inverse of hyperbolic stereographic projection. This is illustrated in the figure to the right for $n = 2$ . It is instructive to compare to stereographic projection for spheres.

Stereographic projection $σ : H n R \to R n$ and its inverse $σ -1 : R n \to H n R$ are given by $\begin{aligned} σ (τ, x) = u & = \frac{R x}{R + τ}, \\ σ^{- 1} (u) = (τ, x) & = (R \frac{R^{2} + | u |^{2}}{R^{2} - | u |^{2}}, \frac{2 R^{2} u}{R^{2} - | u |^{2}}), \end{aligned}$ where, for simplicity, $τ \equiv ct$ . The $(τ, x)$ are coordinates on $M n +1$ and the $u$ are coordinates on $R n$ .

Detailed derivation

Let $H_{R}^{n} = {(τ, x^{1}, \dots, x^{n}) \subset M : - τ^{2} + {(x^{1})}^{2} + \dots + {(x^{n})}^{2} = - R^{2}, τ > 0}$ and let $S = (- R, 0, \dots, 0) .$

If $P = (τ, x^{1}, \dots, x^{n}) \in H_{R}^{n},$ then it is geometrically clear that the vector $\vec{P S}$ intersects the hyperplane ${(τ, x^{1}, \dots, x^{n}) \in M : τ = 0}$ once in point denoted $U = (0, u^{1} (P), \dots, u^{n} (P)) \equiv (0, u) .$

One has $\begin{aligned} S + \vec{S U} & = U \Rightarrow \vec{S U} = U - S, \\ S + \vec{S P} & = P \Rightarrow \vec{S P} = P - S \end{aligned}$ or $\begin{aligned} \vec{S U} & = (0, u) - (- R, 0) = (R, u), \\ \vec{S P} & = (τ, x) - (- R, 0) = (τ + R, x) . \end{aligned} .$

By construction of stereographic projection one has $\vec{S U} = λ (τ) \vec{S P} .$

This leads to the system of equations $\begin{aligned} R & = λ (τ + R), \\ u & = λ x . \end{aligned}$

The first of these is solved for $λ$ and one obtains for stereographic projection $σ (τ, x) = u = \frac{R x}{R + τ} .$

Next, the inverse $σ -1 (u) = (τ, x)$ must be calculated. Use the same considerations as before, but now with $\begin{aligned} U & = (0, u) \\ P & = (τ (u), x (u)) . \end{aligned},$ one gets $\begin{aligned} τ & = \frac{R (1 - λ)}{λ}, \\ x & = \frac{u}{λ}, \end{aligned}$ but now with $λ$ depending on $u$ . The condition for $P$ lying in the hyperboloid is $- τ^{2} + | x |^{2} = - R^{2},$ or $- \frac{R^{2} (1 - λ)^{2}}{λ^{2}} + \frac{| u |^{2}}{λ^{2}} = - R^{2},$ leading to $λ = \frac{R^{2} - | u |^{2}}{2 R^{2}} .$

With this $λ$ , one obtains $σ^{- 1} (u) = (τ, x) = (R \frac{R^{2} + | u |^{2}}{R^{2} - | u |^{2}}, \frac{2 R^{2} u}{R^{2} - | u |^{2}}) .$

Pulling back the metric

One has $h_{R}^{1 (n)} = η |_{H_{R}^{1 (n)}} = {(d x^{1})}^{2} + \dots + {(d x^{n})}^{2} - d τ^{2}$ and the map $σ^{- 1} : R^{n} \to H_{R}^{1 (n)}; σ^{- 1} (u) = (τ (u), x (u)) = (R \frac{R^{2} + | u |^{2}}{R^{2} - | u |^{2}}, \frac{2 R^{2} u}{R^{2} - | u |^{2}}) .$

The pulled back metric can be obtained by straightforward methods of calculus; ${{(σ^{- 1})}^{*} η |}_{H_{R}^{1 (n)}} = {(d x^{1} (u))}^{2} + \dots + {(d x^{n} (u))}^{2} - {(d τ (u))}^{2} .$

One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives), $\begin{aligned} d x^{1} (u) & = d (\frac{2 R^{2} u^{1}}{R^{2} - | u |^{2}}) = \frac{\partial}{\partial u^{1}} \frac{2 R^{2} u^{1}}{R^{2} - | u |^{2}} d u^{1} + \dots + \frac{\partial}{\partial u^{n}} \frac{2 R^{2} u^{1}}{R^{2} - | u |^{2}} d u^{n} + \frac{\partial}{\partial τ} \frac{2 R^{2} u^{1}}{R^{2} - | u |^{2}} d τ, \\ ⋮ \\ d x^{n} (u) & = d (\frac{2 R^{2} u^{n}}{R^{2} - | u |^{2}}) = \dots, \\ d τ (u) & = d (R \frac{R^{2} + | u |^{2}}{R^{2} - | u |^{2}}) = \dots, \end{aligned}$ and substitutes the results into the right hand side. This yields ${(σ^{- 1})}^{*} h_{R}^{1 (n)} = \frac{4 R^{2} [{(d u^{1})}^{2} + \dots + {(d u^{n})}^{2}]}{{(R^{2} - | u |^{2})}^{2}} \equiv h_{R}^{2 (n)} .$

Detailed outline of computation

This last equation shows that the metric on the ball is identical to the Riemannian metric $h 2(n) R$ in the Poincaré ball model, another standard model of hyperbolic geometry.

World

From Wikipedia, the free encyclopedia

The world is the totality of entities, the whole of reality, or everything that exists. The nature of the world has been conceptualized differently in different fields. Some conceptions see the world as unique, while others talk of a "plurality of worlds". Some treat the world as one simple object, while others analyze the world as a complex made up of parts.

In scientific cosmology, the world or universe is commonly defined as "the totality of all space and time; all that is, has been, and will be". Theories of modality talk of possible worlds as complete and consistent ways how things could have been. Phenomenology, starting from the horizon of co-given objects present in the periphery of every experience, defines the world as the biggest horizon, or the "horizon of all horizons". In philosophy of mind, the world is contrasted with the mind as that which is represented by the mind.

Theology conceptualizes the world in relation to God, for example, as God's creation, as identical to God, or as the two being interdependent. In religions, there is a tendency to downgrade the material or sensory world in favor of a spiritual world to be sought through religious practice. A comprehensive representation of the world and our place in it, as is found in religions, is known as a worldview. Cosmogony is the field that studies the origin or creation of the world, while eschatology refers to the science or doctrine of the last things or of the end of the world.

In various contexts, the term "world" takes a more restricted meaning associated, for example, with the Earth and all life on it, with humanity as a whole, or with an international or intercontinental scope. In this sense, world history refers to the history of humanity as a whole, and world politics is the discipline of political science studying issues that transcend nations and continents. Other examples include terms such as "world religion", "world language", "world government", "world war", "world population", "world economy", or "world championship".

Etymology

The English word world comes from the Old English weorold. The Old English is a reflex of the Common Germanic *weraldiz, a compound of weraz 'man' and aldiz 'age', thus literally meaning roughly 'age of man'; this word led to Old Frisian warld, Old Saxon werold, Old Dutch werolt, Old High German weralt, and Old Norse verǫld.

The corresponding word in Latin is mundus, literally 'clean, elegant', itself a loan translation of Greek cosmos 'orderly arrangement'. While the Germanic word thus reflects a mythological notion of a "domain of Man" (compare Midgard), presumably as opposed to the divine sphere on the one hand and the chthonic sphere of the underworld on the other, the Greco-Latin term expresses a notion of creation as an act of establishing order out of chaos.

Conceptions

Different fields often work with quite different conceptions of the essential features associated with the term "world". Some conceptions see the world as unique: there can be no more than one world. Others talk of a "plurality of worlds". Some see worlds as complex things composed of many substances as their parts while others hold that worlds are simple in the sense that there is only one substance: the world as a whole. Some characterize worlds in terms of objective spacetime while others define them relative to the horizon present in each experience. These different characterizations are not always exclusive: it may be possible to combine some without leading to a contradiction. Most of them agree that worlds are unified totalities.

Monism and pluralism

Monism is a thesis about oneness: that only one thing exists in a certain sense. The denial of monism is pluralism, the thesis that, in a certain sense, more than one thing exists. There are many forms of monism and pluralism, but in relation to the world as a whole, two are of special interest: existence monism/pluralism and priority monism/pluralism. Existence monism states that the world is the only concrete object there is. This means that all the concrete "objects" we encounter in our daily lives, including apples, cars and ourselves, are not truly objects in a strict sense. Instead, they are just dependent aspects of the world-object. Such a world-object is simple in the sense that it does not have any genuine parts. For this reason, it has also been referred to as "blobject" since it lacks an internal structure like a blob. Priority monism allows that there are other concrete objects besides the world. But it holds that these objects do not have the most fundamental form of existence, that they somehow depend on the existence of the world. The corresponding forms of pluralism state that the world is complex in the sense that it is made up of concrete, independent objects.

Scientific cosmology

Scientific cosmology can be defined as the science of the universe as a whole. In it, the terms "universe" and "cosmos" are usually used as synonyms for the term "world". One common definition of the world/universe found in this field is as "[t]he totality of all space and time; all that is, has been, and will be". Some definitions emphasize that there are two other aspects to the universe besides spacetime: forms of energy or matter, like stars and particles, and laws of nature. World-conceptions in this field differ both concerning their notion of spacetime and of the contents of spacetime. The theory of relativity plays a central role in modern cosmology and its conception of space and time. A difference from its predecessors is that it conceives space and time not as distinct dimensions but as a single four-dimensional manifold called spacetime. This can be seen in special relativity in relation to the Minkowski metric, which includes both spatial and temporal components in its definition of distance. General relativity goes one step further by integrating the concept of mass into the concept of spacetime as its curvature. Quantum cosmology uses a classical notion of spacetime and conceives the whole world as one big wave function expressing the probability of finding particles in a given location.

Theories of modality

The world-concept plays a role in many modern theories of modality, sometimes in the form of possible worlds. A possible world is a complete and consistent way how things could have been. The actual world is a possible world since the way things are is a way things could have been. There are many other ways things could have been besides how they actually are. For example, Hillary Clinton did not win the 2016 US election, but she could have won them. So there is a possible world in which she did. There is a vast number of possible worlds, one corresponding to each such difference, no matter how small or big, as long as no outright contradictions are introduced this way.

Possible worlds are often conceived as abstract objects, for example, in terms of non-obtaining states of affairs or as maximally consistent sets of propositions. On such a view, they can even be seen as belonging to the actual world. Another way to conceive possible worlds, made famous by David Lewis, is as concrete entities. On this conception, there is no important difference between the actual world and possible worlds: both are conceived as concrete, inclusive and spatiotemporally connected. The only difference is that the actual world is the world we live in, while other possible worlds are not inhabited by us but by our counterparts. Everything within a world is spatiotemporally connected to everything else but the different worlds do not share a common spacetime: They are spatiotemporally isolated from each other. This is what makes them separate worlds.

It has been suggested that, besides possible worlds, there are also impossible worlds. Possible worlds are ways things could have been, so impossible worlds are ways things could not have been. Such worlds involve a contradiction, like a world in which Hillary Clinton both won and lost the 2016 US election. Both possible and impossible worlds have in common the idea that they are totalities of their constituents.

Phenomenology

Within phenomenology, worlds are defined in terms of horizons of experiences. When we perceive an object, like a house, we do not just experience this object at the center of our attention but also various other objects surrounding it, given in the periphery. The term "horizon" refers to these co-given objects, which are usually experienced only in a vague, indeterminate manner. The perception of a house involves various horizons, corresponding to the neighborhood, the city, the country, the Earth, etc. In this context, the world is the biggest horizon or the "horizon of all horizons". It is common among phenomenologists to understand the world not just as a spatiotemporal collection of objects but as additionally incorporating various other relations between these objects. These relations include, for example, indication-relations that help us anticipate one object given the appearances of another object and means-end-relations or functional involvements relevant for practical concerns.

Philosophy of mind

In philosophy of mind, the term "world" is commonly used in contrast to the term "mind" as that which is represented by the mind. This is sometimes expressed by stating that there is a gap between mind and world and that this gap needs to be overcome for representation to be successful. One problem in philosophy of mind is to explain how the mind is able to bridge this gap and to enter into genuine mind-world-relations, for example, in the form of perception, knowledge or action. This is necessary for the world to be able to rationally constrain the activity of the mind. According to a realist position, the world is something distinct and independent from the mind. Idealists conceive of the world as partially or fully determined by the mind. Immanuel Kant's transcendental idealism, for example, posits that the spatiotemporal structure of the world is imposed by the mind on reality but lacks independent existence otherwise. A more radical idealist conception of the world can be found in Berkeley's subjective idealism, which holds that the world as a whole, including all everyday objects like tables, cats, trees and ourselves, "consists of nothing but minds and ideas".

Theology

Different theological positions hold different conceptions of the world based on its relation to God. Classical theism states that God is wholly distinct from the world. But the world depends for its existence on God, both because God created the world and because He maintains or conserves it. This is sometimes understood in analogy to how humans create and conserve ideas in their imagination, with the difference being that the divine mind is vastly more powerful. On such a view, God has absolute, ultimate reality in contrast to the lower ontological status ascribed to the world. God's involvement in the world is often understood along the lines of a personal, benevolent God who looks after and guides His creation. Deists agree with theists that God created the world but deny any subsequent, personal involvement in it. Pantheists reject the separation between God and world. Instead, they claim that the two are identical. This means that there is nothing to the world that does not belong to God and that there is nothing to God beyond what is found in the world. Panentheism constitutes a middle ground between theism and pantheism. Against theism, it holds that God and the world are interrelated and depend on each other. Against pantheism, it holds that there is no outright identity between the two.^[45]

History of philosophy

In philosophy, the term world has several possible meanings. In some contexts, it refers to everything that makes up reality or the physical universe. In others, it can mean have a specific ontological sense (see world disclosure). While clarifying the concept of world has arguably always been among the basic tasks of Western philosophy, this theme appears to have been raised explicitly only at the start of the twentieth century,

Plato

Plato is well known for his theory of forms, which posits the existence of two different worlds: the sensible world and the intelligible world. The sensible world is the world we live in, filled with changing physical things we can see, touch and interact with. The intelligible world is the world of invisible, eternal, changeless forms like goodness, beauty, unity and sameness. Plato ascribes a lower ontological status to the sensible world, which only imitates the world of forms. This is due to the fact that physical things exist only to the extent that they participate in the forms that characterize them, while the forms themselves have an independent manner of existence. In this sense, the sensible world is a mere replication of the perfect exemplars found in the world of forms: it never lives up to the original. In the allegory of the cave, Plato compares the physical things we are familiar with to mere shadows of the real things. But not knowing the difference, the prisoners in the cave mistake the shadows for the real things.

Wittgenstein

Two definitions that were both put forward in the 1920s, however, suggest the range of available opinion. "The world is everything that is the case", wrote Ludwig Wittgenstein in his influential Tractatus Logico-Philosophicus, first published in 1921.

Heidegger

Martin Heidegger, meanwhile, argued that "the surrounding world is different for each of us, and notwithstanding that we move about in a common world".

Eugen Fink

"World" is one of the key terms in Eugen Fink's philosophy. He thinks that there is a misguided tendency in western philosophy to understand the world as one enormously big thing containing all the small everyday things we are familiar with. He sees this view as a form of forgetfulness of the world and tries to oppose it by what he calls the "cosmological difference": the difference between the world and the inner-worldly things it contains. On his view, the world is the totality of the inner-worldly things that transcends them. It is itself groundless but it provides a ground for things. It therefore cannot be identified with a mere container. Instead, the world gives appearance to inner-worldly things, it provides them with a place, a beginning and an end. One difficulty in investigating the world is that we never encounter it since it is not just one more thing that appears to us. This is why Fink uses the notion of play or playing to elucidate the nature of the world. He sees play as a symbol of the world that is both part of it and that represents it. Play usually comes with a form of imaginary play-world involving various things relevant to the play. But just like the play is more than the imaginary realities appearing in it so the world is more than the actual things appearing in it.

Goodman

The concept of worlds plays a central role in Nelson Goodman's late philosophy. He argues that we need to posit different worlds in order to account for the fact that there are different incompatible truths found in reality. Two truths are incompatible if they ascribe incompatible properties to the same thing. This happens, for example, when we assert both that the earth moves and that the earth is at rest. These incompatible truths correspond to two different ways of describing the world: heliocentrism and geocentrism. Goodman terms such descriptions "world versions". He holds a correspondence theory of truth: a world version is true if it corresponds to a world. Incompatible true world versions correspond to different worlds. It is common for theories of modality to posit the existence of a plurality of possible worlds. But Goodman's theory is different since it posits a plurality not of possible but of actual worlds. Such a position is in danger of involving a contradiction: there cannot be a plurality of actual worlds if worlds are defined as maximally inclusive wholes. This danger may be avoided by interpreting Goodman's world-concept not as maximally inclusive wholes in the absolute sense but in relation to its corresponding world-version: a world contains all and only the entities that its world-version describes.

Religion

Yggdrasil, an attempt to reconstruct the Norse world tree which connects the heavens, the world, and the underworld.

Mythological cosmologies depict the world as centered on an axis mundi and delimited by a boundary such as a world ocean, a world serpent or similar.

Hinduism

Hinduism constitutes a family of religious-philosophical views. These views present perspectives on the nature and role of the world. Samkhya philosophy, for example, is a metaphysical dualism that understands reality as comprising 2 parts: purusha and prakriti. The term "purusha" stands for the individual conscious self that each of "us" possesses. Prakriti, on the other hand, is the 1 world inhabited by all these selves. Samkhya understands this world as a world of matter governed by the law of cause and effect. The term "matter" is understood in a sense in this tradition including physical and mental aspects. This is reflected in the doctrine of tattvas, according to which prakriti is made up of 23 principles or elements of reality. These principles include physical elements, like water or earth, and mental aspects, like intelligence or sense-impressions. The relation between purusha and prakriti is conceived as 1 of observation: purusha is the conscious self aware of the world of prakriti and does not causally interact with it.

A conception of the world is present in Advaita Vedanta, the monist school among the Vedanta schools. Unlike the realist position defended in Samkhya philosophy, Advaita Vedanta sees the world of multiplicity as an illusion, referred to as Maya. This illusion includes impression of existing as separate experiencing selfs called Jivas. Instead, Advaita Vedanta teaches that on the most fundamental level of reality, referred to as Brahman, there exists no plurality or difference. All there is is 1 all-encompassing self: Atman. Ignorance is seen as the source of this illusion, which results in bondage to the world of mere appearances. Liberation is possible in the course of overcoming this illusion by acquiring the knowledge of Brahman, according to Advaita Vedanta.

Christianity

Contemptus mundi is the name given to the belief that the world, in all its vanity, is nothing more than a futile attempt to hide from God by stifling our desire for the good and the holy. This view has been characterised as a "pastoral of fear" by historian Jean Delumeau. "The world, the flesh, and the devil" is a traditional division of the sources of temptation.

Orbis Catholicus is a Latin phrase meaning "Catholic world", per the expression Urbi et Orbi, and refers to that area of Christendom under papal supremacy.

Islam

In Islam, the term "dunya" is used for the world. Its meaning is derived from the root word "dana", a term for "near". It is associated with the temporal, sensory world and earthly concerns, i.e. with this world in contrast to the spiritual world. Religious teachings warn of a tendency to seek happiness in this world and advise a more ascetic lifestyle concerned with the afterlife. Other strands in Islam recommend a balanced approach.

Mandaeism

In Mandaean cosmology, the world or earthly realm is known as Tibil. It is separated from the World of Light (alma d-nhūra) above and the World of Darkness (alma d-hšuka) below by aether (ayar).

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